by John Summa (Contact Author  Biography)
After looking at the concept of position Greeks in previous tutorial segments, here we will return to a look at some outright positions in order to reveal how Greeks interrelate, and how their interaction in options strategies can alter the prospects for profit and loss. One key area is highlighted here  how Theta and Delta relate and are impacted by changing levels of implied volatility.
In previous segments the Greeks were largely inspected in isolation (the ceteris paribus assumption). But if we relax the assumption of all other things remaining the same (ceteris paribus), then a more complicated picture emerges in terms of the behavior of Greeks.
Take the example of position Delta. Recall that at different strikes across time the Delta rises with more time premium on the option. Taking a look at Figure 14, we can subject Delta to a fall in implied volatility. Remember, when implied volatility falls, so will extrinsic value on those options. The higher the extrinsic value before the fall, the more risk there is in terms of Vega exposure, so typically higher extrinsic value will mean higher Vega values. Since options have negative Vega because they lose value when implied volatility falls, the size of the loss can be measured by looking at Vega on each option, or if a strategy uses a combination of options, by looking at the position Vega.
Figure 14 provides a look at Deltas assuming no time value decay. The 1410 outofthemoney call strike Delta (righthand column) suffers a significant decline with falling IV (left hand column). This decline is exacerbated when the 8 days have elapsed, as seen in Figure 15. For instance, with no implied volatility changes and no time value decay, the Delta is 38.05, as seen in Figure 14. But with a fall in implied volatility by 4 points (4) and passage of 8 calendar days, the Delta declines to 28.25, as seen in Figure 15, making any long position a more unlikely winner and a short position a more likely winner. Additional declines in implied volatility and time value decay will eventually reduce Delta to single digits as seen, for instance, in Figure 16. Here, with a 10 fall in IV at 17 days into the trade, Delta has fallen to just 8.97.
While the Delta risk is greater for the 1360 call at all time intervals, the Delta decay rate due to falling IV is much lower with the closertothemoney (in this case slightly inthemoney) 1360 call options, as seen in Figures 1417. The severity of the decline in Delta on the 1410 call is revealed in Figures 182 and Tables 2122. The at to inthemoney 1360 maintains its Delta values despite declines in IV and time value. However, because the option has more time premium, there is always more directional risk should the market move the wrong way, a difficult tradeoff, particularly for option buyers. (To learn more about time value, read The Importance of Time Value.)
Conclusion
In this segment, atthemoney and outofthemoney call options are used to contrast the impact on Delta values resulting from changes to levels of implied volatility and time remaining on the options. It is demonstrated that while more potential dollar risk may reside on an atthemoney call option, it does not experience the corrosive effects of falling implied volatility and passage of time that an outofthemoney call option does (a put option would similarly face the same conditions). While outofthemoney options cost less (and therefore have much less maximum risk for a buyer), they display much larger Delta decay risk.
Options Greeks: Conclusion
After looking at the concept of position Greeks in previous tutorial segments, here we will return to a look at some outright positions in order to reveal how Greeks interrelate, and how their interaction in options strategies can alter the prospects for profit and loss. One key area is highlighted here  how Theta and Delta relate and are impacted by changing levels of implied volatility.
In previous segments the Greeks were largely inspected in isolation (the ceteris paribus assumption). But if we relax the assumption of all other things remaining the same (ceteris paribus), then a more complicated picture emerges in terms of the behavior of Greeks.
Take the example of position Delta. Recall that at different strikes across time the Delta rises with more time premium on the option. Taking a look at Figure 14, we can subject Delta to a fall in implied volatility. Remember, when implied volatility falls, so will extrinsic value on those options. The higher the extrinsic value before the fall, the more risk there is in terms of Vega exposure, so typically higher extrinsic value will mean higher Vega values. Since options have negative Vega because they lose value when implied volatility falls, the size of the loss can be measured by looking at Vega on each option, or if a strategy uses a combination of options, by looking at the position Vega.
Simulated Changes to IV Levels 
1360 AttheMoney Call Option Strike  1410 OutoftheMoney Call Option Strike 
Delta Values w/ No Time Value Decay  
0  58.63  38.05 
2  58.55  35.80 
4  58.41  33.15 
6  58.26  29.96 
8  58.11  26.11 
10  57.93  21.40 
Figure 14 : Delta values for S&P 500 futures February call options Deltas and associated changes in implied volatility (IV). S&P 500 futures price at 1361.60. As IV falls, Delta values decline. But the fall in Delta is higher for the at and outofthemoney calls. Assumes no time value decay. 
Figure 14 provides a look at Deltas assuming no time value decay. The 1410 outofthemoney call strike Delta (righthand column) suffers a significant decline with falling IV (left hand column). This decline is exacerbated when the 8 days have elapsed, as seen in Figure 15. For instance, with no implied volatility changes and no time value decay, the Delta is 38.05, as seen in Figure 14. But with a fall in implied volatility by 4 points (4) and passage of 8 calendar days, the Delta declines to 28.25, as seen in Figure 15, making any long position a more unlikely winner and a short position a more likely winner. Additional declines in implied volatility and time value decay will eventually reduce Delta to single digits as seen, for instance, in Figure 16. Here, with a 10 fall in IV at 17 days into the trade, Delta has fallen to just 8.97.
Simulated Changes to IV Levels 
1360 AttheMoney Call Option Strike  1410 OutoftheMoney Call Option Strike 
Delta Values w/ 8 Days in Time Value Decay  
0  57.57  33.60 
2  57.45  31.14 
4  57.32  28.25 
6  57.18  24.84 
8  57.04  20.82 
10  56.87  16.13 
Figure 15: Delta values for S&P 500 futures February call options with 8 calendar days remaining until expiration and associated changes in implied volatility (IV). S&P 500 futures price at 1361.60. As IV falls, Delta values decline. But the fall in Delta is higher for the at and outofthemoney calls. 
Simulated Changes to IV Levels 
1360 AttheMoney Call Option Strike  1410 Out–oftheMoney Call Option Strike 
Delta Values w/ 17 Days in Time Value Decay  
0  56.26  26.44 
2  56.16  23.69 
4  56.00  20.57 
6  55.92  17.03 
8  55.80  13.11 
10  55.60  8.97 
Figure 16: Delta values for S&P 500 futures February call options with 17 calendar days remaining until expiration and associated changes in implied volatility (IV). S&P 500 futures price at 1361.60. As IV falls, Delta values decline. But the fall in Delta is higher for the at and outofthemoney calls. 
Simulated Changes to IV Levels 
1360 AttheMoney Call Option Strike  1410 OutoftheMoney Call Option Strike 
Delta Values w/ 25 Days Time Value Decay  
0  55.20  14.96 
2  55.10  12.11 
4  55.05  9.14 
6  55.00  6.17 
8  54.95  3.5 
10  54.90  1.4 
Figure 17: Delta values for S&P 500 futures February call options with 33 calendar days remaining until expiration and associated changes in implied volatility (IV). S&P 500 futures price at 1361.60. As IV falls, Delta values decline. But the fall in Delta is higher for the at and outofthemoney calls. 42.50 is the price of the 1360 call and 18.90 is the price of the 1410 call. Clearly ultimately, more is at risk with the higher priced call (higher maximum potential loss). 
While the Delta risk is greater for the 1360 call at all time intervals, the Delta decay rate due to falling IV is much lower with the closertothemoney (in this case slightly inthemoney) 1360 call options, as seen in Figures 1417. The severity of the decline in Delta on the 1410 call is revealed in Figures 182 and Tables 2122. The at to inthemoney 1360 maintains its Delta values despite declines in IV and time value. However, because the option has more time premium, there is always more directional risk should the market move the wrong way, a difficult tradeoff, particularly for option buyers. (To learn more about time value, read The Importance of Time Value.)
Figure 18: Delta values at different days remaining. Assumes no change in volatility levels. 
Figure 19: Delta values at different days remaining. Assumes 2% points change in volatility levels. 
Figure 20: Delta values at different days remaining. Assumes 6% points change in IV levels. 
% Point Change in IV  33 Days Left  25 Days Left  16 Days Left  8 Days Left 
0  38.05  33.6  26.44  14.96 
2  35.8  31.14  23.69  12.11 
4  33.15  28.25  20.57  9.14 
6  29.96  24.84  17.03  6.17 
8  26.11  20.82  13.11  3.5 
10  21.4  16.13  8.97  1.4 
Total Rate of Change in Delta  43.76  51.99  66.07  90.64 
Figure 21: Delta values and total rates of decay of Delta for 1410 call options with changing levels of IV and decays remaining on the option. The highest rate of decline in Delta takes place on the option with eight days remaining, which suffers a 90.64% drop in Delta with a 10 fall in IV in terms of percentage points change. 
% Point Change in IV  33 Days Left  25 Days Left  16 Days Left  8 Days Left 
0  58.63  57.57  56.26  55.2 
2  58.55  57.45  56.16  55.1 
4  58.41  57.32  56  55.05 
6  58.26  57.18  55.92  55 
8  58.11  57.04  55.8  54.95 
10  57.93  56.87  55.6  54.9 
Total Rate of Change in Delta  1.1  1.22  1.17  0.54 
Figure 21: Delta values and total rates of decay of Delta for 1360 call options with changing levels of IV and decays remaining on the option. The highest rate of decline in Delta takes place on the option with 33 days remaining, which suffers a 1.19% drop in Delta with a 10 fall in IV in terms of percentage points change. Contrast this with the outofthemoney 1410 call, which suffers the smallest decline in Delta on the 33 day intervals. However, the magnitudes are much greater: a 43.76% drop on the 1410 call versus a 1.19% drop in Delta on the 1360 call. 
Conclusion
In this segment, atthemoney and outofthemoney call options are used to contrast the impact on Delta values resulting from changes to levels of implied volatility and time remaining on the options. It is demonstrated that while more potential dollar risk may reside on an atthemoney call option, it does not experience the corrosive effects of falling implied volatility and passage of time that an outofthemoney call option does (a put option would similarly face the same conditions). While outofthemoney options cost less (and therefore have much less maximum risk for a buyer), they display much larger Delta decay risk.
Options Greeks: Conclusion
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